|
In mathematics, a function〔The words map or mapping, transformation, correspondence, and operator are often used synonymously. .〕 is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number ''x'' to its square ''x''2. The output of a function ''f'' corresponding to an input ''x'' is denoted by ''f''(''x'') (read "''f'' of ''x''"). In this example, if the input is −3, then the output is 9, and we may write ''f''(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write ''f''(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function. Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation. The input and output of a function can be expressed as an ordered pair, ordered so that the first element is the input (or tuple of inputs, if the function takes more than one input), and the second is the output. In the example above, ''f''(''x'') = ''x''2, we have the ordered pair (−3, 9). If both input and output are real numbers, this ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function. In modern mathematics, a function is defined by its set of inputs, called the ''domain''; a set containing the set of outputs, and possibly additional elements, as members, called its ''codomain''; and the set of all input-output pairs, called its ''graph''. Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the ''image'' of the function). For example, we could define a function using the rule ''f''(''x'') = ''x''2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (''x'', ''x''2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, and division of functions, in those cases where the output is a number. Another important operation defined on functions is function composition, where the output from one function becomes the input to another function. ==Introduction and examples== For an example of a function, let ''X'' be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square; and let ''Y'' be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape to its color is a function from ''X'' to ''Y'': each shape is linked to a color (i.e., an element in ''Y''), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no shape that has two or more colors. This function will be referred to as the "color-of-the-shape function". The input to a function is called the argument and the output is called the value. The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the codomain. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does ''not'' require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in ''X''. A second example of a function is the following: the domain is chosen to be the set of natural numbers (1, 2, 3, 4, ...), and the codomain is the set of integers (..., −3, −2, −1, 0, 1, 2, 3, ...). The function associates to any natural number ''n'' the number 4−''n''. For example, to 1 it associates 3 and to 10 it associates −6. A third example of a function has the set of polygons as domain and the set of natural numbers as codomain. The function associates a polygon with its number of vertices. For example, a triangle is associated with the number 3, a square with the number 4, and so on. The term range is sometimes used either for the codomain or for the set of all the actual values a function has. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Function (mathematics)」の詳細全文を読む スポンサード リンク
|